and \(a_k, b_k\)  are arbitrary complex constants. Obviously, the function \(\tau _n=\det _{1\le i,j\le N}\left(\phi _{2i-1,2j-1}^{(n)}\right)\)  with the above choice of \(\phi _{ij}^{(n)}\)  also satisfies the bilinear equations (3.14) in [1]}. Then, when we set \(b_k=a_k^*\) , \(x_1=x\)  and \(x_2=\textrm {i}t/2\) , this \(\tau _n\)  function would satisfy the bilinear equations (3.1) of the NLS equation in [1]} [with \(t\)  switched to \(-t/2\)  since the NLS equation (REF ) in this paper differs from that in [1]} by this \(t\)  rescaling]. Applying the same reduction technique of [1]} to the above new \(\tau _n\)  solution, we can remove the differential operators in the expression (REF ) of its matrix element \(\phi _{ij}^{(n)}\)  and reduce it to \(\sigma _n=\det _{1\le i,j\le N}\left(\phi _{2i-1,2j-1}^{(n)}\right)\) , where
\(\phi _{i,j}^{(n)}=\sum _{\nu =0}^{\min (i,j)} \frac{1}{4^{\nu }} \hspace{1.70709pt} S_{i-\nu }(\hat{\textbf {\emph {x}}}^{+}(n) +\nu \textbf {\emph {s}}) \hspace{1.70709pt} S_{j-\nu }(\hat{\textbf {\emph {x}}}^{-}(n) + \nu \textbf {\emph {s}}),\) 
